Meshless approach based on MLS with additional constraints for large deformation analysis

In the present work, we are interested to develop a meshless approach, based on the strong form MLS approximation with additional constraints, to solve the nonlinear elastic and elsto-plastic problems for regular and irregular distribution of points. We adopt a plastic behavior law based on the total deformation theory, which is very convenient when the physical nonlinearity is more important than the effect of irreversible process and the loading history. In plasticity, one encounters discontinuities of rigidity where the application of asymptotic developments seems difficult or impossible. To apply the Taylor series expansion, regularization methods have been adapted. The strong form MLS approximation allows us to avoid the inconvenient of the numerical integration, while the asymptotic developments help us to reduce the computation cost observed in the incremental law of plasticity and the iterative methods. For irregular points distribution, we can get an ill-posed least squares problems due to a singular moment matrix of MLS approximation. To avoid this difficulty, we propose a modified MLS approximation by introducing additional constraints which allows to increase the error functional used in the derivation of the shape functions.