## 2 A Numerical Solution Developed Using the Collocation Point Method
### A Dislocation near a Cylindrical Void
The critical distance to avoid any effect of the different but statically equivalent loads/force distributions should be equal to or greater than the average spacing between the collocation points (\(q\)) [23; 24]. This--that is, the application of [PERSON]'s principle--is shown clearly for meshes with 16 and 32 elements. The results shown in the plots of Figure 9 can also be shown by inversely varying the D for different \(q\) values (i.e., different rectangular mesh densities). According to Figure 10, the effect of cylindrical void meshing with different numbers of rectangular dislocation loops (i.e., with different stress fields that are statically equivalent) dies out when the distance D is greater than \(q\). In other words, the critical distance D must be larger than \(q\) to ensure the accuracy of the numerical solution, which is consistent with statements in [23; 24]. Figure 9: Comparison plots between the analytical solution and numerical results (rectangular elements, D = \(2b\), \(r_{0}=4\sqrt{2}b\), \(l\) = 1 million \(b\), \(\mathbf{\xi}=100b\), \(\overset{\rightarrow}{b}\) for the real dislocation, the infinite dislocation or the driving force is \((1,0,0)\)) (**a**) normalized \(\sigma_{xx}\) (**b**) normalized \(\sigma_{yy}\)). It is important to note that the rectangular elements are increased along the circumferential direction (see Figures 5b and 6), and all collocation points in this case are located on the \(xy\)-plane (\(z=0\)). The average spacing between collocation points (\(q\)) for this uniform mesh is the same here as the distance between the collocation points (\(s\)) along the circumferential direction. For the triangular elements, \(q\) is much larger than \(s\) as the centers of the triangular dislocation loops/meshing elements, or the collocation points, are far away from the \(xy\)-plane, and the distance between the collocation points of two neighboring triangular elements (distance measured along the longitudinal direction or the length of the cylinder) is close to infinity for the almost infinite cylinder considered here (see Figure 5a and imagine the cylinder is infinitely long). Therefore, the critical distance D to ensure the accuracy of the numerical solution developed by the collocation point method is expected to be larger than the distance \(s\) between the collocation points along the circumferential direction. This is shown in Figure 11. Although triangular elements or triangular dislocation loops offer more flexibility for irregular shapes or curved surfaces, their centers or collocation points are far from the \(xy\)-plane and hence from the plotted field points. As the centers of the rectangular elements (or their collocation points) are all on the \(xy\)-plane (at \(z=0\)), this provides a better numerical solution compared to the 2D analytical solution in this scenario. It should be recalled here that the stress field points are plotted on the \(xy\)-plane, \(z=0\). Figure 11 also shows that this issue can be overcome with distance D versus \(q\) by simply increasing the number of elements or the mesh density (i.e., by simply decreasing \(s\)). Figure 10: Comparison plots between the analytical solution and numerical results for normalized \(\sigma_{xy}\) (rectangular elements, \(r_{0}=4\sqrt{2}b\), \(l=1\) million \(b\), \(\xi=100b\), \(b\) for the real dislocation, the infinite dislocation or the driving force is (1, 0, 0)) (**a**) D = 2\(b\) (**b**) D = 3\(b\) (**c**) D = 5\(b\) (**d**) D = 8\(b\)). Before concluding this section, it is worth noting the effect of the length of the cylinder (\(l\)) on the numerical solution. In studies of the effects of mesh density and the critical distance D, \(l\) is maintained constant at 1 million \(b\), which is a number close to infinity compared with the radius of the cylinder \(r_{0}=4\sqrt{2}b\), in order to verify our numerical solution with the 2D analytical solution.