In this study, a numerical meshless method is used to solve the weak form of the linear elastic equations in solid mechanics. Evaluation and comparison of the numerical meshless methods have been carried out via the radial point interpolation meshless method with multi-quadrics base functions (MQ-RPIM) and meshless local Petrov-Galerkin method (MLPG). Using these two methods, stress intensity factors in an elastic medium containing geometric discontinuities and cracks are estimated based on tensile and bending cyclic loading. The analysis domain has been identified via three-dimensional modeling of the notched and un-notched shafts with an initial surface semi-elliptical crack subjected to tensile or bending cyclic loadings. To enhance the accuracy of calculations, the RPIM meshless method is applied using polynomial and extended-enriched 3D base functions. Shape functions have been developed using standard and optimal parameters and values with Mono-Objective Function in PSO algorithm. In the MLPG meshless method with the extended-enriched functions, discretization is performed via direct and penalty factor methods, to reach more efficient results and meet the boundary conditions. Efficiency comparison of the selected numerical methods with the experimental findings and the numerical analysis of finite elements method indicates that in comparison with the MLPG method, MQ-RPIM enriched meshless method can be utilized with fewer nodes in the analysis domain while reaching the accuracy and convergence with lower stress intensity factors and gentler slope. However, the processing time of the MLPG meshless method is lower than that of the other methods.

本研究采用数值无网格方法求解固体力学中线弹性方程的弱形式。数值无网格方法的评估和比较通过径向点插值无网格方法与多二次基函数（MQ-RPIM）和无网格局部彼得罗夫-伽辽金方法（MLPG）进行。使用这两种方法，基于拉伸和弯曲循环载荷估计包含几何不连续性和裂纹的弹性介质中的应力强度因子。分析域已通过对具有初始表面半椭圆裂纹的有缺口和无缺口轴进行三维建模来确定，该轴受到拉伸或弯曲循环载荷。为了提高计算的准确性，RPIM 无网格方法使用多项式和扩展丰富的 3D 基函数来应用。形状函数是使用标准和最优参数和值在 PSO 算法中使用单目标函数开发的。在具有扩展丰富函数的MLPG无网格方法中，通过直接和惩罚因子方法进行离散化，以获得更有效的结果并满足边界条件。所选数值方法与实验结果和有限元方法数值分析的效率比较表明，与MLPG方法相比，MQ-RPIM丰富的无网格方法可以在分析域中使用更少的节点，同时达到精度和收敛性具有较低的应力强度因子和较缓的坡度。然而，