## 2 A Numerical Solution Developed Using the Collocation Point Method
### A Dislocation near a Cylindrical Void
In the investigation of the effects of \(l,32\) rectangular elements or rectangular dislocation loops are used to uniformly mesh the curved surface of the cylinder. All other dimensional parameters are kept unchanged. The stress fields are plotted on the \(xy\)-plane along the line parallel to the \(y\)-axis and shifted D = \(2b\) distance away from the right edge of the cylinder along the \(x\)-axis. As shown in Figure 12, the mismatch between the numerical results and the 2D analytical solution diminishes as the length of the cylinder (\(l\)) increases. Eventually, the numerical solution and the analytical solution overlap as \(l\) approaches relative infinity (i.e., 1 million \(b\)) compared with \(r_{0}=4\sqrt{2}b\). According to the Saint-Venant's principle, the numerical solution should have matched the analytical solution as the distance D (D = \(2b\)) is greater than the spacing between the neighboring collocation points (\(q=s=1b\)). However, the length of the cylinder (\(l\)) is another key factor affecting the accuracy of the numerical solution. When \(l\) is too small, say \(l=4b\), the numerical result cannot match the analytical solution even though D is greater than \(q\). Figure 11: Comparison plots between the analytical solution and numerical results for normalized \(\sigma_{xx}\) (triangular elements, \(s=4.5b\) or \(1.63b\), \(r_{0}=4\sqrt{2}b\), \(l=1\) million \(b\), \(q\)-\(l\), \(\xi=100b\), \(\overset{\rightarrow}{b}\) for the real dislocation, the infinite dislocation or the driving force is (1, 0, 0)) (**a**) D = \(2b\) (**b**) D = \(5b\) (**c**) D = \(10b\) (**d**) D = \(20b\). (see Figure 12), as the 2D analytical solution is for the stress field of an infinite edge dislocation outside an infinitely long cylindrical void. Conversely, the numerical solution and the analytical result are in perfect agreement when \(l\) is sufficiently large, as demonstrated in Figure 12.  
Overall, verification of the numerical solution developed by the collocation point method for the stress fields of a dislocation near a cylindrical void has been achieved. The numerical solution and the 2D analytical solution match without any deviation as long as the critical distance D is larger than the average spacing between collocation points (\(q\)), and the cylindrical void is infinitely long compared with the radius of the cylinder \(r_{0}\). For the general 3D numerical solution, the real dislocation or the driving force can be any type of dislocation (e.g., a dislocation loop or a dislocation segment), and the cylindrical void can be infinitely long or of finite length or of any shape and not just cylindrical. To ensure the accuracy of the 3D numerical solution, the mesh needs to be sufficiently dense and the Saint-Venant's principle is applied. The authors are not familiar with any fully-3D analytical dislocation-void problems with which to directly compare these results, except for the 2D analytical solution provided above. Theory of elasticity-based dislocation solutions are limited to field points beyond a core radius (usually between \(b\) and \(4b\)[8]) of a dislocation (whether a crystal dislocation or a mesh dislocation). The 3D numerical solution can be implemented into three-dimensional dislocation dynamics codes for dynamic simulations to investigate the interaction between a dislocation source and a void of any shape, and the mechanical plastic properties (e.g., the flow stress) of materials with such voids. Figure 8:0 Comparison plots between the analytical solution and numerical results for normalized \(\sigma_{xx}\) (32 rectangular elements, D = 2\(b\), \(q=s=1b\), \(r_{0}=4\sqrt{2}b\), \(l=1\) million \(b\), \(\xi=100b\), \(\overrightarrow{b}\) for the real dislocation, the infinite dislocation or the driving force is (1, 0, 0)). This plot shows the effect of cylinder length on the stress results.