An advanced meshless technique for large deformation analysis of metal forming

The large deformation analysis is one of major challenges in numerical modelling and simulation of metal forming. Although the finite element method (FEM) is a well-established method for modeling nonlinear problems, it often encounters difficulties for large deformation analyses due to the mesh distortion issues. Because no mesh is used, the meshless methods show very good potential for the large deformation analysis. In this paper, a local meshless formulation is developed for the large deformation analysis. The Radial Basis Function (RBF) is employed to construct the meshless shape functions, and the spline function with high continuity is used as the weight function in the construction of the local weak form. The discrete equations for large deformation of solids are obtained using the local weak-forms, RBF shape functions, and the total Lagrangian (TL) approach, which refers all variables to the initial (undeformed) configuration. This formulation requires no explicit mesh in computation and therefore fully avoids mesh distortion difficulties in the large deformation analysis of metal forming. Several example problems are presented to demonstrate the effectiveness of the developed meshless technique. It has been found that the developed meshless technique provides a superior performance to the conventional FEM in dealing with large deformation problems in metal forming

A semi-proximal-based strictly contractive Peaceman-Rachford splitting method

The Peaceman-Rachford splitting method is very efficient for minimizing sum of two functions each depends on its variable, and the constraint is a linear equality. However, its convergence was not guaranteed without extra requirements. Very recently, He et al. (SIAM J. Optim. 24: 1011 - 1040, 2014) proved the convergence of a strictly contractive Peaceman-Rachford splitting method by employing a suitable underdetermined relaxation factor. In this paper, we further extend the so-called strictly contractive Peaceman-Rachford splitting method by using two different relaxation factors, and to make the method more flexible, we introduce semi-proximal terms to the subproblems. We characterize the relation of these two factors, and show that one factor is always underdetermined while the other one is allowed to be larger than 1. Such a flexible conditions makes it possible to cover the Glowinski's ADMM whith larger stepsize. We show that the proposed modified strictly contractive Peaceman-Rachford splitting method is convergent and also prove $O(1/t)$ convergence rate in ergodic and nonergodic sense, respectively. The numerical tests on an extensive collection of problems demonstrate the efficiency of the proposed method