A Boundary Enhancement for the Stabilized Conforming Nodal Integration of Galerkin Meshfree Methods

The stabilized conforming nodal integration (SCNI) has been successfully developed for Galerkin meshfree methods based upon the linear exactness requirement. In this study, it is shown that for a given problem domain, when the support of the meshfree shape functions associated with the interior nodes do not cover the essential boundary, the linear exactness can be perfectly achieved by the standard SCNI formulation. On the other hand, when the essential boundary lies in the support of the meshfree shape functions of the interior nodes, a linear field may not be exactly obtained with the original SCNI formulation where the essential boundary conditions are enforced via the nodally exact transformation method, and the error even becomes more pronounced with the increase of support size. To resolve this issue, a flux term associated with the essential boundary is recovered in the variational formulation and it turns out to be proper to keep this term since the meshfree shape functions of interior nodes usually do not vanish on the boundary. Consequently the original SCNI integration constraint is revised and the stiffness matrix is enhanced by an additional stiffness contribution from the flux integration along the essential boundary. It is demonstrated that the proposed enhanced formulation is capable of exactly reproducing linear fields regardless of the support sizes. Moreover, several benchmark examples reveal that the present SCNI formulation with boundary enhancement yields better accuracy compared with the original SCNI approach, particularly for meshfree discretizations with larger support sizes.

Efficient Meshfree Formulation for Metal Forming Simulations

A stabilized conforming (SC) nodal integration method is developed for elastoplastic contact analysis of metal forming processes. In this approach, strain smoothing stabilization is introduced to eliminate spatial instability in Galerkin meshfree methods using nodal integration. The gradient matrix associated with strain smoothing satisfies the integration constraint (IC) of linear exactness in the Galerkin approximation. Strain smoothing formulation and numerical procedures for history-dependent problems are introduced. Applications to metal forming analysis are presented, with the results demonstrating a significant improvement in computational efficiency without loss of accuracy.

Efficient Meshfree Formulation for Metal Forming Simulations

A stabilized conforming (SC) nodal integration method is developed for elastoplastic contact analysis of metal forming processes. In this approach, strain smoothing stabilization is introduced to eliminate spatial instability in collocation meshfree methods. The gradient matrix associated with strain smoothing satisfies the integration constraint (IC) of linear exactness in the Galerkin approximation. Strain smoothing formulation and numerical procedures for history-dependent problems are introduced. Applications to metal forming analysis are presented, with the results demonstrating a significant improvement in computational efficiency without loss of accuracy.