三维摩擦接触问题算法精度和收敛性研究

Study of convergence of exact algorithm to solve the three-dimensional frictional contact problems

回顾了摩擦接触问题的现有解法,包括Lagrange乘子法、惩罚函数法、增广Lagrangian乘子法,线性互补模型及互补类非线性方程组方法等,尤其是近期关于非线性方程组方法的系列研究结果:(1)三维弹性摩擦接触问题互补类非线性方程组的光滑化解法;(2)将其推广为非光滑解法;(3)三维弹塑性摩擦接触问题增量方程和算法.非线性方程组方法解法基于严格的数学理论基础,严格地满足接触条件,并用随机数产生的接触柔度矩阵证实了算法的收敛性,由于每个接触点对的未知数只有3个,不含任何人工变量,同时,算法有局部二次收敛率且计算效率很高.为了考察以增广拉格朗日法为代表的工程中流行算法的精度和收敛性,设计了一个典型的弹性摩擦接触问题算例,证实了通用程序ANSYS的增广拉格朗日法是近似的并且不能保证收敛.

The reviews of the conventional methods including the Lagrange methods, the penalty function methods, the augmented Lagrangian method and the linear complementarity method, and specially, the smooth nonlinear complementarity equations method and the non-smooth nonlinear complementarity equations method (NNCEM) for solving the three-dimensional elastic frictional contact problem are given. It is shown that the contact conditions are satisfied rigorously in NNCEM, and the convergence property is verified numerically by using the randomly generated contact flexibility matrix. The NNEQM is applied to establish an incremental formulation and solution method for solving three-dimensional static elastoplastic frictional contact problems. A typical numerical example is designed to compare the accuracy and convergence property of the NNCEM with that of the augmented Lagrange method used in the commercial finite element code, ANSYS version 5.7. The numerical results show that the NNEQM is more accurate than that employed by ANSYS and the convergence of the NNEQM is guaranteed. Whereas the widely used Augmented Lagrange method is an approximate method, and its convergence is not guaranteed.