A stabilized semi-Lagrangian Galerkin meshfree formulation for extremely large deformation analysis

The Lagrangian meshfree discretization that considers evaluation of kernels based on particle distance measured in the undeformed configuration is ineffective in modeling extremely large deformation due to the regularity requirement of deformation gradient. This research proposes a semi-Lagrangian discretization which performs kernel evaluation using distance measure defined on the deformed configuration. This approach allows the neighbors to be redefined in the deformation process, and avoids the need for inverse mapping from deformed to undeformed configurations. In the proposed semi-Lagrangian formulation, a kernel convective effect resulting from the material diffusion has been derived and an iterative method for updating nodal mass and density is proposed.; A stabilized non-conforming nodal integration (SNNI) has been introduced for integrating Galerkin weak form discretized by the semi-Lagrangian shape functions. SNNI relaxes the integration constraints imposed on the stabilized conforming nodal integration (SCNI) (Chen et al., 2001) and simplifies the strain smoothing procedures for semi-Lagrangian discretization. The results from numerical simulations show that the dispersion and accuracy of SNNI are comparable to those of SCNI. As an enhancement of stability in SCNI/SNNI, a sub-domain integration method has been developed to further stabilize the domain integration for transient problems.; von Neumann stability analyses for Lagrangian and semi-Lagrangian discrete equations integrated using SCNI, SNNI and sub-domain integration methods have been performed. Numerical studies showed that the stability properties of semi-Lagrangian discretization are similar to those of Lagrangian formulation. The results imply that the convective effect in semi-Lagrangian discretization is a consistent treatment of the non-conservative semi-Lagrangian kernels.; Based on the proposed semi-Lagrangian formulations, new particle contact algorithms have been developed for extremely large deformation and self-contact problems. Unlike traditional contact algorithms, the new contact algorithms do not require any contact surfaces to be defined a priori, and they are particularly effective for problems with evolving contact surface topology.