## 3 Numerical Examples
### T-Butt Welded Joint  
The consistency and effectiveness of the improved RPIM-based notch stress solution for T-butt welded joint are presented here. The T-butt welded joint is shown in Figure 11a. The thickness of the T-butt welded joint is 12 mm (\(t~{}=~{}12\) mm). The model is considered as plane strain model. The remote loading \(F~{}=~{}1200\) N was enforced on the right end of this T-butt welded joint. The detailed geometry conditions are shown in Figure 11a. The stress solution \(\sigma_{x}\) at notch location of the T-butt welded joint obtained from conventional RPIM are presented in Figure 11b. As can be seen, stress solutions at weld toe obtained from conventional RPIM are sensitive to the parameters in RPIM, which makes it difficult to apply stress solution of the conventional RPIM in calculation SCFs or stress intensity factors. The SCFs based on the stress solutions of improved methods are plotted in Figure 11c. The mode I stress intensity factor for edge cracks obtained by Equation (49) was plotted in Figure 11d. The notch stress  
Figure 9: Investigation on sensitivity of stress solution to number of field nodes.  
intensities results of the same T-butt welded joint using mesh-insensitive structural stress method by [PERSON] [17] and weight functions by [PERSON] citeB46-applsci-934788 were employed here as validations. It is noted that a small radius existed at notch root of weld in [46], and the radius was used for reducing singularity at welds. The corresponding notch stress intensities factor results are also shown in Figure 11d. The results show that there are good agreements among results based on solution of improved RPIM and results from [17; 46].  
Figure 11: SCFs and notch stress intensities results of T-butt welded joint: (**a**) model definition; (**b**) stress solutions obtained from conventional RPIM; (**c**) results of SCFs based on solutions from improved RPIM and improved EFG; (**d**) validations of notch stress intensities results based on improved RPIM and improved EFG (the improved RPIM-based K was obtained from Equation (49)).  
Figure 10: Comparisons of stress concentration factors (SCFs) obtained from different types of model: (**a**) model definition of the plate lap filter weld (**b**) comparisons of SCFs based on improved RPIM, improved EFG and the mesh-insensitive structural stress method.  
It can be seen from the first example that the stress solution of the conventional RPIM and EFG method are affected by the parameters used even if there is no stress concentration effect. The sensitivity of the stress solution increases obviously when there is existing singularity at the notch. The second and third examples demonstrated that the SCFs and stress intensity factors at the weld toe can be computed consistently and effectively based on stress solution of improved RPIM. Actually, the basic principle of the improved RPIM is that the discrete equations in the form of Equation (28) must satisfy mechanical equilibrium conditions regardless of shape parameters, dimension of the influence domain or number of field nodes. Since the equilibrium conditions are directly enforced on each field node, the most accurate solution variables in conventional RPIM are filed nodes forces. In improved RPIM, the stress solutions in the form of membrane stress and bending stress are obtained directly from field nodes forces in order to avoid using derivatives of shape functions. It is this reason that guarantees the insensitivity of the stress solutions of the improved RPIM.