基于单元速度泰勒展开的上限原理有限元法

Upper bound limit analysis based on Taylor expansion form of element velocity

极限分析上限法较极限平衡法有着严谨的理论基础和物理意义。借助于级数展开的思想,通过对速度在单元形心点处一阶泰勒展开,得到了以形心点速度和速度一阶导数为基本未知变量的上限有限元法,该方法丰富了上限有限元基本理论,且可更好地表达与刚体上限法之间的联系。在形成新的上限法的同时,放松单元内任一点均需严格满足上限性质的要求,采用等面积多边形的形式代替外切多边形来逼近摩尔-库仑屈服圆。算例表明:该方法可稳定收敛到真实解,具有和传统Sloan法相同的收敛度;采用等面积多边形逼近形式时,较少的多边形边数即可取得较好的收敛效果,收敛速度大大提高。

The upper bound limit analysis method has a more rigorous theoretical basis and clearer physics meaning compared to the limit equilibrium method. Based on the series expansion, the velocity field of a triangle element is expanded through center point velocity and its velocity gradients. Hence the upper bound FEM method based on the center point velocity and its velocity gradients are introduced as unknowns. The method not only enriches the upper bound limit analysis method, but also has a simpler flow equation. The proposed method can be considered as the multidimensional extension of rigid FE upper bound limit method, and has a more rigorous theoretical basis than rigid FE upper bound limit method. The requirement that all the point must strictly meet the limit properties is relaxed and the equal-area polygon is adopted to approximate the yield circle. Two case studies show that this method can steadily converge into the true solution, and has the same convergence as the traditional method by Sloan; when equal-area polygon is taken, the method yields a good result by less number of polygon edges, with significantly improved convergence.