Application Of Meshless Local Petrov-Galerkin Method In Large Deformation Problems

Because of its mesh-dependency the finite element method (FEM) is known to have difficulties in solving the problems with large deformation and moving boundaries. Some well-developed meshless Galerkin methods, such as EFG and RKPM, can solve such problems, but they need background cells to conduct numerical integration, and are inefficient in solving large-scale problems. The meshless local Petrov-Galerkin (MLPG) method has the advantage of"truly meshless", high efficiency and accuracy. In this paper some numerical schemes for simulating the large deformation of nonlinear materials are presented in the frame of MLPG and verified by numerical examples. The main contributions are:1. The computational efficiency and numerical stability of the MLPG have been analyzed. Based on analyzing the efficiency of the MLPG computation, including the way of nodal query, matrix storage and equation solving, an optimized scheme is presented to make MLPG an O(N) algorithm. A spectral factoring approach is applied to analyze the numerical stability of MLPG, and the parameters which make MLPG numerically stable is obtained.2. A material configuration based numerical scheme of MLPG for the large deformation analysis of path-independent materials is presented, and the incremental formulation is obtained with conventional linearization approach. Some numerical examples of hyperelastic material are given to verify the presented numerical scheme, and the results indicate that the nonlinear MLPG method is capable to ease volume locking without any other treatment, and it can simulate much larger deformation than FEM and avoid the drawback of FEM on inaccuracy caused by mesh distortion.3. A numerical scheme of MLPG for the large deformation analysis of path-dependent materials is presented. In order to avoid the shape updating of trial and test subdomains, variables are represented in the spatial coordinate ofcurrent configuration, while the numerical quadrature is conducted in the material coordinate of initial configuration, and the derivatives of shape functions with respect to the spatial coordinate are converted to material coordinate by tensor transformation. A multiplicative constitutive model of hyperelastoplasticity is adopted to simulate large deformation. To verify the presented numerical scheme, it is used to simulate some typical strain localization phenomenon, such as necking and shear-band forming. The numerical examples have shown that the presented MLPG method is capable to simulate large plastic deformation accurately, and it avoids volume locking caused by volume preserving of plastic strain and mesh distortion by large deformation. Furthermore, the meshless approximation function of MLPG can reproduce high order displacement field, thus it can ease the numerical difficulties induced by the discontinuity line of shear band.