## 2 A Numerical Solution Developed Using the Collocation Point Method
### A Dislocation near a Cylindrical Void
On the contrary, the analytical solution is developed for a 2D problem, i.e., the stress solution of an infinite edge dislocation near an infinitely long cylindrical void or a circular hole. The length of the cylindrical void in our numerical solution needs to be close to infinite or relatively infinite compared with the radius of the cylinder, in order to make the comparison between the numerical results and the 2D analytical solution. Additionally, an infinite dislocation should be used as the real dislocation or the driving force in our numerical solution to match the analytical results. Before discussing and verifying the results, some parameters are specified and symbolized for simplicity, as follows: Shear modulus: G; Poisson's ratio: \(\
u\); Burger's vector: \(\overrightarrow{b}=(b_{x},b_{y},b_{z})\); radius of the cylinder: \(r_{0}\); length of the cylinder: \(l\); location of the dislocation: \((\xi,0)\); coordinate of the cylinder's center: \((0,0)\); distance between the right edge of the cylinder and the field points line along the \(y\)-axis, as shown in Figure 8: D; average spacing between the collocation points: \(q\); and distance between the neighboring collocation points along the circumferential direction \(s\), as shown in Figure 5.  
On a separate note, the dimensional parameters \(\xi\), D, \(l\), \(q\), and \(s\) are all normalized by the magnitude of the Burger's vector \(b\), and the stress is normalized by the shear modulus. The purpose of normalization is to transform parameters and results at different scales to be on a similar scale, and to improve the generalization of the current numerical solution [28,29]. For the comparison plots, the stress field of a dislocation near an infinitely long cylindrical void is plotted on the \(xy\)-plane (\(z=0\)) along a line parallel to the \(y\)-axis and shifted D distance away from the right edge of the cylinder along the \(x\)-axis; see the dashed green line in Figure 8. Defining the following parameters for plotting: G = 110 GPa, \(\
u=0.36\) (elastic properties of copper [30]), \(r_{0}=4\sqrt{2}b\), \(l=1\) million \(b\), \(\xi=100b\); \(\overrightarrow{b}\) for the real dislocation, the infinite dislocation, or the driving force, is \((1,0,0)\). Figure 9 shows that the numerical solution approaches the analytical solution [19,27] as the number of rectangular dislocation loops padding the curved surface of the cylinder increases. In numerical methods, the solution is approximated by essentially piecewise functions over each element. More elements allow for a better approximation of the geometry and the numerical solution can change more precisely from one element to another. Furthermore, the spacing between the elements gets closer as the number of  
Figure 8: Schematic figure for an infinite edge dislocation near a cylindrical hole with the radius \(r_{0}\). elements increases, resulting in a reduction in truncation errors and a more accurate numerical solution [25; 31]. Figure 9 shows the results for field points 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. In this case, the field points are close to the cylindrical void. [PERSON]'s principle states that the elastic fields resulting from different but statically equivalent force distributions on a body are approximately the same far away from the loading points or these force distributions [32]. That being said, the stress field caused by the cylindrical void with different mesh densities should be the same if the field points are far away from the void. The purpose of the stress field induced by the dislocation loops meshing the surface of the void is to annul (at the free-surface collocation points) the stress field arising from the real dislocation, which is constant in this study; hence, the stress fields caused by the cylindrical void with different mesh densities should be statically equivalent.