On the Boundary Conditions in Out-of-Plane Analysis of Thin Plates by the Finite Point Method

The finite point method (FPM) is a numerical, mesh-free technique for solving differential equations, particularly in fluid dynamics. While the FPM has been previously applied in solid mechanics to analyze plates under in-plane loading, there remains a notable scarcity of research exploring the out-of-plane analysis of elastic plates using this method. This study thoroughly investigates the elastic FPM analysis of thin plates subjected to transverse loadings, focusing specifically on various boundary conditions (BCs). Boundary conditions represent a significant challenge in the out-of-plane analysis of thin plates within the FPM framework. To address this challenge, the approach incorporates additional nodal points positioned close to each boundary node, supplementing the points distributed throughout the plate's interior and along its edges. The strong form of the governing equation is employed for the interior points, while the analysis also includes the scenario of a plate resting on boundary columns. Both distributed and concentrated external loads are examined to provide a comprehensive understanding of the behavior under different loading conditions. Furthermore, the optimal placement of the extra boundary nodes is briefly discussed, alongside a focus on the number of nodes within the finite point clouds. An appropriate range for this number is proposed, although the determination of the optimal distance for the extra boundary nodes and the ideal number of cloud points is earmarked for future research. The contribution of this work is to enhance the understanding of the FPM in the context of thin plates, particularly concerning the critical influence of boundary conditions.