基于可行弧内点算法的上限有限单元法优化求解

赵明华; 张锐; 雷勇

doi:10.11779/CJGE201404002

基于可行弧内点算法的上限有限单元法优化求解 English Version

湖南大学岩土工程研究所,湖南长沙 410082

基金项目: 国家自然科学基金项目（51278187）

详细信息

作者简介:
赵明华（1956- ）,男,湖南洞口县人,教授,博士生导师,主要从事桩基及软土地基处理研究。E-mail: mhzhaohd@21cn.com。

中图分类号: TU43；O24
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出版历程
- 收稿日期: 2013-03-26
- 发布日期: 2014-04-21

Optimization of upper bound finite element method based on feasible arc interior point algorithm

Geotechnical Engineering Institute of Hunan University, Changsha 410082, China

摘要

摘要: 上限有限单元法将寻找机动相容速度场的问题转化为一个数学规划问题,克服了人为构造机动相容速度场的困难,在复杂工程问题中具有广阔的应用前景。基于非线性规划的上限有限单元法,可避免对屈服函数的线性化处理,大大地减少了优化变量数,同时可节约大量存储空间,但由此产生的非线性规划模型十分复杂。为此,在引入一种非线性上限规划模型的基础上,探讨基于可行弧内点算法对其进行优化求解的步骤。首先,采用BFGS公式对屈服函数的Hessian矩阵进行迭代,避免了计算过程中该矩阵病态的问题；其次,通过构造可行弧,克服了当迭代点到达非线性约束边界时搜索步长过短的问题；最后,采用Wolfe非精确搜索技术进行线性搜索,提高了步长搜索效率。通过MATLAB编程进行算例分析表明,基于可行弧内点算法的非线性上限有限单元法,计算效率高、计算误差小、数值稳定性好,可以适应大部分土体稳定性分析计算。
- 极限分析上限法 /
- 有限元单元法 /
- 非线性规划 /
- 可行弧内点算法
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.
- upper bound limit analysis /
- finite element method /
- nonlinear programming /
- feasible arc interior point algorithm

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参考文献(19)

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