非连续变形分析方法变分不等式提法的外梯度法

Extra-gradient method for the variational inequality formulation of discontinuous deformation analysis

在非连续变形分析法(DDA)中块体之间的接触约束是通过罚函数法实现的,罚乘子的选择是否合理将直接影响到数值计算能否顺利进行和计算结果是否可信.本项研究将接触力作为独立未知量,采用变分不等式描述典型"角-边"格式接触在切向和法向上必须满足的接触条件,将DDA重构为一个变分不等式问题,然后采用外梯度法(extra-gradient method)进行求解,在避开引入罚乘子的同时也无须进行"开-闭迭代",其表达形式紧凑,求解计算过程可避开大规模非线性方程组的求解,只需进行投影操作,通过算例的应用表明该方案是可行的、且可更高精度地满足接触条件.

In the conventional discontinuous deformation analysis（DDA） method the contact conditions are enforced by the penalty function method. Improperly selected penalty parameters might incur numerical instability. In order to evade the introduction of the penalty parameters and to avoid ＂open-close iteration＂ that can not assure convergence, in this study the contact forces are taken as independent variables, the normal and tangential contact conditions of one angle-edge contact are expressed as two variational inequalities, then DDA is reformulated as a variational inequality problem which is solved by the extra-gradient method. The proposed computation scheme is more compact and robust, the solution of large-scale nonlinear equations is avoided and only simple projection is executed in the new scheme. Some typical examples originally designed by Shi are reanalyzed, suggesting the effectiveness of the Extra-Gradient Method and contact conditions could be satisfied more strictly.