Enhancing displacement accuracy of stable node-based smoothed finite element method (SNS-FEM) with a smoothing framework based on weighted least squares (WLS)

The Node-based Smoothed Finite Element Method (NS-FEM) mitigates strain and volumetric locking in the standard Finite Element Method (FEM) by employing averaged nodal strains and nodal integration to construct the stiffness matrix. However, nodal integration may fail to constrain certain distortion modes, resulting in numerical instability. Stable NS-FEM (SNS-FEM) introduces temporary integration points that constrain these spurious modes, restoring stability. This paper proposes a new formulation that enhances the displacement accuracy of SNS-FEM, particularly on coarse meshes. Instead of relying on local strain integration and averaging, the method computes nodal strains using a Weighted Least Squares (WLS) approach. By leveraging the flexibility of WLS, it strategically selects strain sampling locations and polynomial fitting orders to modify the computed nodal strain gradients. Compared to SNS-FEM, this modification leads to a lower stiffness contribution from the stabilisation term, thereby improving displacement accuracy. The formulation is evaluated across four types of test cases, focusing primarily on displacement fields and natural frequency modes. Results demonstrate that the proposed method preserves dynamic stability while improving displacement accuracy relative to SNS-FEM—especially on coarse meshes—at comparable computational costs.