A stabilized one-point integrated mixed formulation for finite element and meshfree methods in modeling nearly incompressible materials

This study develops a stabilized Galerkin mixed formulation within a one-point integrated framework to model nearly incompressible materials. The variational multiscale formulation addresses the hourglass modes and Ladyzhenskaya–Babusska–Brezzi (LBB) instability in conventional Galerkin mixed formulation. The multiscale decomposition of the variational equation results in a residual-based stabilized Galerkin framework. The derived fine-scaled terms act as a stabilization that alleviates hourglass instability and pressure oscillations. The strain smoothing method is employed in coarse-scale terms to satisfy the integration constraint. A smoothed divergence is applied to the fine-scale terms to relax using a higher-order basis function. The proposed method is applied to the reproducing kernel particle method and the smoothed finite element method to demonstrate its efficacy in general meshfree and mesh-based methods. The effectiveness of the formulation is verified by solving a series of elasticity problems under the incompressible limit. The proposed method outperforms the other two classical stabilization approaches, which are also derived in this study.