广义最优准则法研究

 利用函数的二阶Taylor级数展开,将原问题转化为一系列的近似问题,并基于Kuhn-Tucker条件,直接给出了一种新的设计迭代式。该迭代式被证明是二次收敛的。本文所提方法可同时处理各种不同类型约束的结构优化问题,具有广义最优准则法的意义。在结构优化领域,灵敏度分析是重要的研究内容之一。本文利用广义虚载荷概念,建立了应力、位移响应的一阶和二阶导数计算公式。该导数公式对弹性矩阵未加限制,且适用于元素内应力非常值的元素。 本文处理了具有几何约束、应力约束和位移约束的结构优化设计问题。数值结果表明,此方法的计算效率很高,一般仅需2～3次分析迭代便可达工程所需的精度要求,具有实用价值。

ON A GENERALIZED OPTIMALITY CRITERIA METHOD

By using the second order Taylor series expansion, the original problem of structural optimization is reduced to a sequence of approximate problems. On the basis of the Kuhn-Tucher conditions, two iterative expressions for determining design variables and Lagrangian Multipliers are obtained. The convergence of this method is proved mathematically to be of the second order. The method can be used to solve problems with various kinds of behavioral constraints as long as the second Taylor series expansions of the functions involved in the given problem can be generated, and therefore it is of generalized and unified significance in theory and practice.Sensitivity analysis plays an important role in structural optimization. Based on the pseudo load concepts, general formulas for computing the first and second derivatives of behavioral parameters with respect to design variables are derived. By applying the Saint-Venant's principle to structures, the Hessian matrices are reduced to diagonal matrices, and the computational efficiency of design modification is enhanced accordingly.Several typical structures with stress, displacement and side constraints have been optimized to illustrate the power of this method. The results of the computations show that the method presented in this paper is quite efficient, and in general only two or three iterations are needed to meet the engineering accuracy requirement.