Non-linear version of stabilized conforming nodal integration for Galerkin mesh-free methods

A stabilized conforming (SC) nodal integration, which meets the integration constraint in the Galerkin mesh-free approximation, is generalized for non-linear problems. Using a Lagrangian discretization, the integration constraints for SC nodal integration are imposed in the undeformed con guration. This is accomplished by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal integration in the undeformed con guration. The proposed method is independent to the path dependency of the materials. An assumed strain method is employed to formulate the discrete equilibrium equations, and the smoothed deformation gradient serves as the stabilization mechanism in the nodally integrated variational equation. Eigenvalue analysis demonstrated that the proposed strain smoothing provides a stabilization to the nodally integrated discrete equations. By employing Lagrangian shape functions, the computation of smoothed gradient matrix for deformation gradient is only necessary in the initial stage, and it can be stored and reused in the subsequent load steps. A signi cant gain in computational e ciency is achieved, as well as enhanced accuracy, in comparison with the mesh-free solution using Gauss integration. The performance of the proposed method is shown to be quite robust in dealing with non-uniform discretization