# A Solution Procedure to Improve 3D Solid Finite Element Analysis with an Enrichment Scheme
## 2 Finite Element Formulation Enriched by Cover Functions
\begin{bmatrix}Q_{x\overline{x}}^{2}&Q_{x\overline{ y}}^{2}&Q_{x\overline{x}}^{2}&Q_{x\overline{y}}Q_{x\overline{y}}&Q_{x\overline{y}}Q_{x \overline{z}}&Q_{x\overline{z}}Q_{x\overline{x}}\\ Q_{y\overline{x}}^{2}&Q_{y\overline{y}}^{2}&Q_{y\overline{z}}^{2}&Q_{y \overline{x}}\Updelta_{y\overline{y}}&Q_{y\overline{y}}Q_{y\overline{z}}&Q_{y \overline{z}}Q_{y\overline{x}}\\ Q_{z\overline{x}}^{2}&Q_{z\overline{y}}^{2}&Q_{z\overline{z}}^{2}&Q_{z \overline{x}}\Updelta_{z\overline{y}}&Q_{z\overline{y}}Q_{z\overline{y}}&Q_{z \overline{y}}Q_{z\overline{y}}&Q_{z\overline{y}}Q_{z\overline{x}}&Q_{z \overline{z}}Q_{z\overline{x}}\\ 2Q_{x\overline{x}}\Updelta_{y\overline{x}}&2Q_{x\overline{y}}\Updelta_{y \overline{z}}&2Q_{x\overline{y}}\Updelta_{z\overline{z}}&Q_{x\overline{y}} \Updelta_{z\overline{y}}+Q_{x\overline{y}}\Updelta_{y\overline{z}}&Q_{x \overline{y}}Q_{y\overline{z}}+Q_{x\overline{x}}\Updelta_{y\overline{y}}&Q_{x \overline{z}}Q_{y\overline{x}}+Q_{x\overline{x}}\Updelta_{y\overline{z}}\\ 2Q_{y\overline{x}}\Updelta_{z\overline{y}}&2Q_{y\overline{y}}Q_{z\overline{ z}}&2Q_{x\overline{y}}\Updelta_{z\overline{y}}+Q_{y\overline{y}}\Updelta_{z\overline{x}}&Q_{y \overline{z}}Q_{z\overline{x}}+Q_{y\overline{z}}\Updelta_{y\overline{z}}&Q_{y \overline{z}}Q_{z\overline{x}}+Q_{y\overline{y}}\Updelta_{z\overline{x}}\\ 2Q_{x\overline{y}}Q_{x\overline{x}}&2Q_{z\overline{y}}Q_{x\overline{y}}&2Q_{z \overline{z}}Q_{x\overline{z}}&Q_{z\overline{x}}Q_{x\overline{y}}+Q_{x \overline{y}}\Updelta_{x\overline{y}}&Q_{x\overline{z}}\Updelta_{x \overline{z}}+Q_{z\overline{x}}\Updelta_{x\overline{y}}&Q_{z\overline{z}} \Updelta_{x\overline{x}}+Q_{z\overline{x}}\Updelta_{x\overline{z}}\\ \end{bmatrix} , \tag{10}\]  
in which \(Q_{x\overline{y}}=\mathbf{e}_{x}\cdot\overline{\mathbf{e}}_{y}\) is the dot product of \(\mathbf{e}_{x}\) and \(\overline{\mathbf{e}}_{y}\). Here, \(\mathbf{e}_{x}\) and \(\overline{\mathbf{e}}_{y}\) are the unit vectors in the \(x\)-direction of the global Cartesian coordinate system and in the \(y\)-direction of the local Cartesian coordinate system, respectively. See Figure 1a. For isotropic material, the stress vector in the global Cartesian coordinate system is written by  
\[\boldsymbol{\sigma}= \begin{bmatrix}\sigma_{xx}&\sigma_{yy}&\sigma_{zz}&\sigma _{xy}&\sigma_{yz}&\sigma_{zx}\end{bmatrix} ^{\mathrm{T}}=\mathbf{C}^{\mathrm{ iso}}\varepsilon, \tag{11}\]  
with  
\[\mathbf{C}^{\mathrm{iso}}=\frac{E}{(1+\
u)(1-2\
u)} \begin{bmatrix}1-\
u&\
u&\
u& 0&0&0\\ \
u&1-\
u&\
u&0&0&0\\ \
u&\
u&1-\
u&0&0&0\\ 0&0&0&(1-2\
u)/2&0&0\\ 0&0&0&0&(1-2\
u)/2&0\\ 0&0&0&0&0&(1-2\
u)/2\end{bmatrix} , \tag{12}\]  
where \(E\) and \(\
u\) are the elastic modulus and Poisson's ratio, respectively. For orthotropic material, the stress vector in the local Cartesian coordinate system is given as  
\[\overline{\sigma}= \begin{bmatrix}\sigma_{\overline{xx}}&\sigma_{\overline{yy}}& \sigma_{\overline{zz}}&\sigma_{\overline{yy}}&\sigma_{\overline{yz}}&\sigma_{ \overline{zz}}\end{bmatrix} ^{\mathrm{T}}=\overline{\mathbf{C}}^{\mathrm{ orth}}\overline{\varepsilon}, \tag{13}\]with  
\[\overline{\mathbf{C}}^{\mathrm{ortho}}=\left[