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SUMMARYA 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 conÿguration. This is accomplished by introducing a Lagrangian strain smoothing to the deformation gradient, and by performing a nodal integration in the undeformed conÿguration. 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 signiÿcant 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.

SUMMARYA stabilized conforming nodal integration scheme is implemented in the natural neighbour method in conjunction with non-Sibsonian interpolation. In this approach, both the shape functions and the integration scheme are defined through use of first-order Voronoi diagrams. The method illustrates improved performance and significant advantages over previous natural neighbour formulations. The method also shows substantial promise for problems with large deformations and for the computation of higher-order gradients.

A Meshless approach based on a Reproducing Kernel Particle Method is developed for metal forming analysis. In this approach, the displacement shape functions are constructed using the reproducing kernel approximation that satis®es consistency conditions. The variational equation of materials with loading-path dependent behavior and contact conditions is formulated with reference to the current con®guration. A Lagrangian kernel function, and its corresponding reproducing kernel shape function, are constructed using material coordinates for the Lagrangian discretization of the variational equation. The spatial derivatives of the Lagrangian reproducing kernel shape functions involved in the stress computation of path-dependent materials are performed by an inverse mapping that requires the inversion of the deformation gradient. A collocation formulation is used in the discretization of the boundary integral of the contact constraint equations formulated by a penalty method. By the use of a transformation method, the contact constraints are imposed directly on the contact nodes, and consequently the contact forces and their associated stiffness matrices are formulated at the nodal coordinate. Numerical examples are given to verify the accuracy of the proposed meshless method for metal forming analysis.

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