Meshless Local Petrov-Galerkin (MLPG) Approaches for Solving Nonlinear Problems with Large Deformations and Rotations

Abstract

A nonlinear formulation of the Meshless Local Petrov-Galerkin (MLPG) finite-volume mixed method is developed for the large deformation analysis of static and dynamic problems. In the present MLPG large deformation formulation, the velocity gradients are interpolated independently, to avoid the time consuming differentiations of the shape functions at all integration points. The nodal values of velocity gradients are expressed in terms of the independently interpolated nodal values of displacements (or velocities), by enforcing the compatibility conditions directly at the nodal points. For validating the present large deformation MLPG formulation, two example problems are considered: 1) large deformations and rotations of a hyper-elastic cantilever beam, and 2) impact of an elastic-plastic solid rod (cylinder) on a rigid surface (often called as the Taylor impact test). The MLPG result for the cantilever beam problem was successfully compared with results from both analytical modeling and a commercial finite element code simulation. The final shapes of the plastically deformed rod obtained from a well-known finite element code, and the present MLPG code were also successfully compared. The direct comparison of computer run times between the finite element method (FEM) and the large deformation mixed MLPG method showed that the MLPG method was relatively more efficient than the FEM, at least for the two example problems considered in the present study.