Stabilized conforming nodal integration for Galerkin meshfree methods

Domain integration by Gauss quadrature in the Galerkin meshfree methods adds considerable complexity to solution procedures. Direct nodal integration, on the other hand, leads to a numerical instability due to under integration and vanishing derivatives of shape functions at the nodes. A strain smoothing stabilization for nodal integration is proposed to eliminate spatial instability in nodal integration. Eigenvalue analysis demonstrated that the proposed strain smoothing provides stabilization to the nodally integrated discrete equations. No numerical control parameter is involved in the proposed strain smoothing stabilization.; For convergence, an integration constraint (IC) is introduced as a necessary condition for a linear exactness in the meshfree Galerkin approximation. The gradient matrix of strain smoothing is shown to satisfy IC using a divergence theorem. The numerical results show that the accuracy and convergent rates in the meshfree method with a direct nodal integration are improved considerably by the proposed stabilized conforming nodal integration method. It is also demonstrated that the Gauss integration method fails to meet IC in meshfree discretization. For this reason, the proposed method provides even better accuracy than meshfree solution using Gauss integration as presented in several numerical examples.; The proposed stabilized conforming (SC) nodal integration is generalized for nonlinear problems. Using a Lagrangian discretization, the integration constraints for SC nodal integration are imposed in the undeformed configuration. This is accomplished by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal integration in the undeformed configuration. 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. 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 significant gain in computational efficiency is achieved, as well as enhanced accuracy, in comparison with the meshfree solution using Gauss integration. The performance of the proposed method is shown quite robust in dealing with irregular discretization.