This paper aims to systematically assess the local radial basis function collocation method, structured with multiquadrics (MQs) and polyharmonic splines (PHSs), for solving steady and transient diffusion problems. The boundary value test involves a rectangle with Dirichlet, Neuman, and Robin boundary conditions, and the initial value test is associated with the Dirichlet jump problem on a square. The spectra of the free parameters of the method, i.e., node density, timestep, shape parameter, etc., are analyzed in terms of the average error. It is found that the use of MQs is less stable compared to PHSs for irregular node arrangements. For MQs, the most suitable shape parameter is determined for multiple cases. The relationship of the shape parameter with the total number of nodes, average error, node scattering factor, and the number of nodes in the local subdomain is also provided. For regular node arrangements, MQs produce slightly more accurate results, while for irregular node arrangements, PHSs provide higher accuracy than MQs. PHSs are recommended for use in diffusion problems that require irregular node spacing.

This study employs the fourth-order phase-field method (PFM) to investigate crack propagation. The PFM incurs significant computational costs due to its need for a highly dense node arrangement for accurate crack propagation. This study proposes an adaptive loading step size strategy combined with a scattered node (SCN_var) arrangement with variable spacings. The mechanical and phase-field models are solved using the strong-form meshless local radial basis function collocation method in a staggered approach. The method’s performance is evaluated based on accuracy and computational cost, using regular nodes (RGN) and scattered nodes (SCN_uni) with uniform spacing, as well as SCN_var with variable node spacing. Two benchmark tests are used to analyze the proposed method: a symmetric double-notch tension and a single-edge notch shear test. The analysis shows that the adaptive step size strategy improves numerical stability while the SCN_var significantly reduces computational cost. Using SCN_var, the CPU time is decreased by about thirty times compared to uniform nodes in the tensile case and by approximately three times in the shear case, without sacrificing accuracy. This confirms that directing computational resources to critical regions can significantly reduce CPU time, suggesting that adaptive node redistribution could further enhance computational performance.