Development of three-dimensional least-squares finite element technique for inelastic deformation analysis.

摘要

In this research, a three-dimensional finite element technique to analyze inelastic deformation of porous metallic solids is presented. The finite element technique is based on a least-squares variational principle. There are several unique features to the least-squares technique which sets it apart from the conventional finite element method based on the Galerkin-displacement and other mixed formulations. One feature is that the technique always yields a symmetric and positive-definite system of algebraic equations. Second, the technique is not sensitive to the character of the governing equations, thus the same problem-solving procedure can be used for problems governed by mixed elliptic-hyperbolic equations such as shear banding problems. Another feature is that the mixed least-squares technique circumvents the LBB condition imposed on saddle-point formulations such as the mixed Galerkin procedure.; In the present formulation, deformation is viewed as a continuous process and an incremental approach is adopted for the analysis. The three velocity and six Cauchy stress rate components constitute the set of nodal dependent variables. Linear interpolation functions are used for their finite element approximation. Due to the nature of three-dimensional modeling and the nine degrees of freedom per node, the resulting system coefficient matrix is extremely sparse. A sparse data storage scheme is implemented to minimize the required computer memory size. Since the least-squares technique always generates a symmetric and positive-definite system of equations, an effective preconditioned conjugate gradient iterative solver is chosen to solve the linear system. The deformation and stress histories are determined by simple explicit time integration schemes. A comprehensive set of benchmark problems is treated to explore the performance of the least-squares technique and to verify the computer program. Furthermore, two large-scale inelastic problems are investigated involving the necking behavior of a flat rectangular titanium alloy tensile specimen and the open-die upset forging of a porous aluminum rectangular block. The results obtained are physically realistic and provided insights into finite deformation processes.