## 2 A Numerical Solution Developed Using the Collocation Point Method
### A Dislocation near a Cubic Void
\begin{array}{ll}\sum\limits_{j=1}^{N}\sigma_{xz}^{j\to i}=-\sigma_{xz}^{EF \to i},&\quad i=1,\;2,\ldots N\\ \sum\limits_{j=1}^{N}\sigma_{yz}^{j\to i}=-\sigma_{yz}^{EF\to i},&\quad i=1,\;2, \ldots N\\ \sum\limits_{j=1}^{N}\sigma_{zz}^{j\to i}=-\sigma_{zz}^{EF\to i},&\quad i=1,\;2, \ldots N\end{array}  \tag{4}\]  
where \(\sigma_{\alpha\beta}^{j\to i}\) is the \(\alpha\beta\) stress component of loop \(j\) evaluated at collocation point \(i\) or at the center of loop \(i\). Furthermore, \(\sigma_{\alpha\beta}^{EF\to i}\) is the stress component \(\alpha\beta\) of segment \(EF\) calculated at point \(i\). Stress at any field point for a rectangular dislocation loop in the global coordinate system is provided in the following equation:  
\[\mathbf{\sigma}^{loop}=\mathbf{\sigma}^{AB}+\mathbf{\sigma}^{BC}+\mathbf{\sigma}^{CD}+\mathbf{ \sigma}^{DA} \tag{5}\]  
where \(\mathbf{\sigma}^{AB}\) is the stress due to segment AB of the dislocation loop ABCD, and so forth. One can analyze the stress solution of a dislocation segment [26] and find that the stress solution is linear with respect to the Burgers vector components \(b_{x}\), \(b_{y}\), and \(b_{z}\). This allows the following equation to be inserted into Equation (4):  
\[\sigma_{\alpha\beta}^{j\to i}(x,y,z)=K_{x\alpha\beta}^{j\to i}(x,y,z)b_{x}^{j}+K _{y\alpha\beta}^{j\to i}(x,y,z)b_{y}^{j}+K_{z\alpha\beta}^{j\to i}(x,y,z)b_{z} ^{j} \tag{6}\]  
where \(\sigma_{\alpha\beta}^{j\to i}(x,y,z)\) is the \(\alpha\beta\) stress component for loop \(j\) (or dislocation \(j\) in general) at collocation point \(i\), \(b_{x}^{j}\) is the \(x\)-component of the Burgers vector for the generally prismatic dislocation loop \(j\), and \(K_{x\alpha\beta}^{j\to i}(x,y,z)\) is the kernel term associated with the \(x\)-component of the Burgers vector and evaluated at collocation point \(i\). The kernels \(K_{x\alpha\beta},\;K_{y\alpha\beta}\), and \(K_{z\alpha\beta}\) can be derived from the work of [PERSON] [26], for a dislocation line segment, and are clearly demonstrated in [24]. Hence, derivation of the kernel terms is not shown here for brevity. Figure 2: Cubic void surface meshed with rectangular elements or triangular elements. Inserting Equation (6) into Equation (4), the following system of equations ensues:  
\[ \begin{array}{llll}\sum\limits_{j=1}^{N}\ (b_{x}^{j}\ K_{xxz}^{j \to i}+b_{y}^{j}\ K_{yxz}^{j\to i}+b_{z}^{j}\ K_{xxz}^{j\to i})=-\sigma_{xz}^{EF \to i},&\qquad i=1,\ 2,\ldots N\\ \sum\limits_{j=1}^{N}\ (b_{x}^{j}\ K_{xyz}^{j\to i}+b_{y}^{j}\ K_{yzz}^{j \to i}+b_{z}^{j}\ K_{zyz}^{j\to i})=-\sigma_{yz}^{EF \to i},&\qquad i=1,\ 2,\ldots N\\ \sum\limits_{j=1}^{N}\ (b_{x}^{j}\ K_{zzz}^{j\to i}+b_{y}^{j}\ K_{yzz}^{j \to i}+b_{z}^{j}\ K_{zzz}^{j\to i})=-\sigma_{zz}^{EF \to i},&\qquad i=1,\ 2,\ldots N\\ \end{array}  \tag{7}\]  
The above system represents a system of \(3N\) linear algebraic equations. The unknowns in this system of equations are the Burgers vector components of the \(N\) loops. Equation (7) can be expanded and written as:  
\[ \begin{array}{llll}\left[\begin{array}{ccccc}K_{zzz}^{1 \to i}&K_{yzz}^{1 \to i}&K_{zzz}^{1\to i}&K_{yzz}^{1\to i}&K_{xzz}^{N\to i}\\ K_{yzz}^{1\to i}&K_{yzz}^{1\to i}&K_{yzz}^{1\to i}&\cdots&K_{yzz}^{N \to i}&K_{yzz}^{N\to i}\\ K_{zzz}^{1\to i}&K_{yzz}^{1\to i}&K_{zzz}^{1\to i}&K_{zzz}^{N \to i}&K_{yzz}^{N\to i}&K_{zzz}^{N\to i}\\ \vdots&&\vdots&&\vdots&\vdots\\ K_{zzz}^{1\to N}&K_{yzz}^{1\to N}&K_{zzz}^{1\to N}&K_{xzz}^{N \to N}&K_{yzz}^{N\to N}&K_{xzz}^{N\to N}\\ K_{zzz}^{1\to N}&K_{yzz}^{1\to N}&K_{zzz}^{1\to N}&\cdots&K_{yzz}^{N \to N}&K_{yzz}^{N\to N}&K_{yzz}^{N\to N}\\ K_{zzz}^{1\to N}&K_{zzz}^{1\to N}&K_{zzz}^{1\to N}&K_{xzz}^{N \to N}&K_{yzz}^{N\to N}&K_{yzz}^{N\to N}\\ \end{array}\right]\left\{\begin{array}{ll}b_{x}^{1}\\ b_{y}^{1}\\ b_{y}^{1}\\ b_{z}^{1}\\ b_{z}^{1}\\ b_{z}^{1}\\ b_{z}^{1}\\ b_{z}^{1}\\ b_{z}^{1}\\ b_{z}^{1}\\ b_{z}^{1}\\ b_{z}^{1}\\ b_{x}^{1}\\ b_{x}^{1}\\ \end{array}\right]\\ \end{array}  \tag{8}\]  
Here, the size of the kernel matrix is \(3N\times 3N\).