Abstract：The upper bound finite element method converts the problem of finding a kinematic admissible velocity field into a mathematical programming one, which can overcome the difficulty of artificially constructing a kinematic velocity field, thus, it has a broad prospect in applications to complex problems. The formulation of the upper bound finite element method based on nonlinear programming can avoid linearization of yield functions, as a result, it greatly reduces the optimization variables and saves a great deal of memory space. However, this leads to a nonlinear programming model that is quite complex. By introducing a nonlinear upper bound programming model, the steps for its optimization using feasible arc interior point algorithm are discussed. Firstly, the BFGS formula is taken as the updating rules for Hessian of yield functions to avoid the ill-conditioning problem in computation. Secondly, by constructing a feasible arc, the shortcoming of a too short step when the current iteration point reaches the nonlinear constraint boundary is overcome. Finally, the Wolfe's line search technique is used for step-length search which enhances the line search efficiency. Example analysis by MATLAB programming shows that the proposed method is highly efficient, numerically stable and accurate enough for engineering practice, thus, it is applicable to most soil stability problems.
赵明华, 张锐, 雷勇. 基于可行弧内点算法的上限有限单元法优化求解[J]. 岩土工程学报, 2014, 36(4): 604-611.
ZHAO Ming-hua, ZHANG Rui, LEI Yong. Optimization of upper bound finite element method based on feasible arc interior point algorithm. Chinese J. Geot. Eng., 2014, 36(4): 604-611.
陈祖煜. 土力学经典问题的极限分析上、下限解[J]. 岩土工程学报, 2002, 24(1): 1-11. (CHEN Zu-yu. Limit analysis for the classic problems of soil mechanics[J]. Chinese Journal of Geotechnical Engineering, 2002, 24(1): 1-11. (in Chinese)) CHEN W F. Limit analysis and soil plasticity[M]. Amsterdam: Elsevier Scientific Publishing Company, 1975. ANDERHEGGEN E, KNOPFEL H. Finite element limit analysis using linear programming[J]. International Journal of Solids and Structures, 1972, 8(12): 1413-1431. BOTTERO A, NEGRE R, PASTOR J, et al. Finite element method and limit analysis theory for soil mechanics problems[J]. Computer Methods in Applied Mechanics and Engineering, 1980, 22(1): 131-149. SLOAN S W, KLEEMAN P W. Upper bound limit analysis using discontinuous velocity fields[J]. Computer Methods in Applied Mechanics and Engineering, 1995, 127: 293-314. 杨峰, 阳军生, 张学民. 基于线性规划模型的极限分析上限有限元的实现[J]. 岩土力学, 2011, 32(3): 914-921. (YANG Feng, YANG Jun-sheng, ZHANG Xue-min. Implementation of finite element upper bound solution oflimit analysis based on linear programming model[J]. Rock and Soil Mechanics, 2011, 32(3): 914-921. (in Chinese)) 王均星, 王汉辉, 吴雅峰. 土坡稳定的有限元塑性极限分析上限法研究[J]. 岩石力学与工程学报, 2004, 23(11): 1867-1873. (WANG Jun-xing, WANG Han-hui, WU Ya-feng. Stability analysis of soil slope by finite element method with plastic limit upper bound[J]. Chinese Journal of Rock Mechanics and Engineering, 2004, 23(11): 1867-1873. (in Chinese)) 杨小礼, 李亮, 刘宝琛. 大规模优化及其在上限定理有限元中的应用[J]. 岩土工程学报, 2001, 23(5): 602-605. (YANG Xiao-li, LI Liang, LIU Bao-chen. Large-scale optimization and its application to upper bound theorem using kinematical element method[J]. Chinese Journal of Geotechnical Engineering, 2001, 23(5): 602-605. (in Chinese)) 姜功良. 浅埋软土隧道稳定性极限分析[J]. 土木工程学报, 1998, 31(5): 65-72. (JIANG Gong-liang. Limit analysis of the stability of shallow tunnels in soft ground[J]. China Civil Engineering Journal, 1998, 31(5): 65-72. (in Chinese)) LYAMIN A V, SLOAN S W. Upper bound limit analysis using linear finite elements and non-linear programming[J]. International Journal for Numerical and Analytical Methods in Geomechanics, 2002, 26: 573-611. HERSKOVITS J, SANTOS G. Feasible arc interior point algorithm for nonlinear optimization[C]// Computational Mechanics, New Trends and Applications. CIMNE, Barcelona, 1998. HERSKOVITS J, MAPPA P, GOULART E, et al. Mathematical programming model and algorithms for engineering design optimization[J]. Computer Methods in Applied Mechanics and Engineering, 2005, 194(33): 3244-3268. COHN M Z, MAIER G. Engineering plasticity by mathematical programming[M]. New York: Pergamon Press, 1979. NOCEDAL J, WRIGHT J W. Numerical optimization[M]. New York: Springer, 2006. RAO S S. Engineering optimization: theory and practice[M]. New Jersey: John Wiley & Sons, 2009. ABBO A J, SLOAN S W. A smooth hyperbolic approximation to the Mohr-Coulomb yield criterion[J]. Computers and Structures, 1995, 54(3): 427-441. LYAMIN A V, SLOAN S W. Mesh generation for lower bound limit analysis[J]. Advances in Engineering Software, 2003, 34: 321-338. DUFF I S. A code for the solution of sparse symmetric definite and indefinite systems[J]. ACM Transactions on Mathematical Software, 2004, 30(2): 118-144. 钱家欢, 殷宗泽. 土工原理与计算[M]. 2版. 北京: 中国水利水电出版社, 1996. (QIAN Jia-huan, YIN Zong-ze. Principles of soil engineering and calculation[M]. 2nd ed. Beijing: China Water Power Press, 1996. (in Chinese))