一种二阶锥线性化方法在上限有限元中的应用研究

Investigation and application of a second-order cone linearizing method to finite element upper bound solution

对于平面应变条件下岩土稳定性分析,基于线性规划的上限有限元需对常用的摩尔-库仑屈服准则形成的二阶锥约束进行线性化,直接地处理方法是以外接多边形替代锥体投影形成的圆域。为了提高线性化精度往往需直接增加外接多边形边数,从而造成线性规划模型中决策变量包含大量的塑性乘子变量,使计算难度大为增加甚至变得不可行。为此,引入Ben-Tal和Nemirovsky提出的一种二阶锥线性化方法,并将其嵌入到自编的上限有限元程序。经算例分析发现,该法与外接多边形线性化方法所获计算结果相互印证,且其通过适量的增加决策变量和等式约束数目,能保证摩尔-库仑屈服准则线性化精度,同时形成的线性规划规模更小,可望应用于基于线性规划模型的上限有限元中。

For solving stability problems of soil-mass under plane strain condition,finite element upper bound solution based on linear programming needs to linearize second-order cone constraint,forming by Mohr-Coulomb（M-C） yield criterion.The commonly used method is to replace the circular region projected from cone by external polygon.In order to improve accuracy of linearization,the direct way is to increase external polygon edges.This will result in the decision variables of linear programming model containing a large number of plastic multiplier variables,and leading to increase difficulties or even infeasible of calculating process.In order to solve the problem,a second-order cone linearizing method proposed by Ben-Tal and Nemirovsky is introduced and has been embedded into the coded finite element upper bound program.It is found that the results obtained by Ben-Tal method and results obtained from external linear polygon method are confirming each other after case study.Moreover,Ben-Tal method can improve the accuracy of linearization of M-C yield criterion effectively by appropriately increasing the number of decision variables and equality constraints.Simultaneously,it has proved that the formation of linear programming is small-scale.It is expected to be applied commonly to finite element upper bound solution based on linear programming model.