边坡稳定梯度法优化计算中新的差分格式与收敛准则

A New Difference Scheme and Convergence Criterion for Gradient Optimization Method for Slope Stability

摘要: 用极限平衡方法计算边坡稳定的安全系数，需要采用优化方法寻找最危险滑裂面。梯度法是一种精确的优化方法方法，然而在边坡稳定分析中存在不能准确找到最危险滑裂面的问题。改进了梯度优化方法，提出了优化方向向量的下降差分格式。它解决了采用中心差分格式出现的搜索方向错误的问题，在有效性、计算精度和优化时间方面都优于中心差分格式。指出了传统梯度优化方法中忽视收敛标准的问题，并提出了一个单变量或多变量沿着优化方向进行优化搜索时的收敛准则。采用2阶段圆弧滑动面搜索方法，对3个测试算例的安全系数进行了下降差分格式和中心差分格式优化搜索的对比，表明改进的梯度法结合下降差分格式是能跳出局部极值的精确有效的方法。算例中采用下降差分格式搜索结果的误差均小于给定误差标准，表明所提出的收敛准则是合适的。

Abstract: It is necessary to use optimization methods to find the most dangerous sliding surface for the safety factor of slope stability calculated by the limit equilibrium method. Gradient method is an accurate optimization method, however there may fail to find accurately the most dangerous sliding surface. An improved gradient optimization method with a descent difference scheme for the calculation of the direction vector is proposed. The descent difference scheme is superior to the central difference scheme both in accuracy and consuming time. The problem of wrong search direction occurring in the central difference scheme is dissolved in this scheme. The problem of convergence criterion used in the classical optimization method based on the gradient of the objective function is pointed out. A new convergence criterion for single variable optimization or for multivariable optimization along the gradient direction is proposed. The stability of three test examples is analyzed with a two-stage search method for circular slip surface. The gradient method combined with the descent difference scheme is an accurate and efficient method with an ability of avoiding to fall to a local minimum in a search process. The errors of search results in the test examples are less than the given convergence error. The proposed convergence criterion is appropriate.