Generalized high-order node-based smoothed polygonal fem on arbitrary polygonal meshes for engineering applications

In this study, we propose three novel high-order smoothed finite element method models based on a smoothed derivative extraction principle to reconstruct strain fields for the analysis of linear elastic solid mechanics problems. First, we developed an innovative smoothing-domain discretization strategy. This approach enables the universal implementation of node-based smoothing domains on arbitrary polygonal background meshes, including both convex and concave polygons. Then, the radial point interpolation method is applied to derive shape functions for polygonal elements with arbitrary node configurations. Next, gradient smoothing technique is used in the point interpolation model, requiring only the values of the shape functions on the smoothing domain boundaries, rather than their derivatives. By utilizing gradient smoothing techniques, only the values of the shape functions on the smoothing domain boundaries are required, rather than their derivatives, thereby reducing the continuity requirements for the shape functions and avoiding coordinate mapping. Furthermore, based on a smoothed derivative extraction principle and the alpha-smoothed finite element method, three high-order smoothed finite element models are established: one with linear strain, one with second-order strain, and one combining the two. Finally, a series of classical benchmark examples are used to verify the accuracy, convergence, and stability of the proposed models. Additionally, the method is applied to complex and challenging engineering problems, demonstrating its effectiveness adaptability.