结构拓扑优化应力敏度分析的伴随法

Adjoint method for stress sensitivity analysis of structural topology optimization

针对受应力约束的连续体结构拓扑优化问题,推导了应力敏度分析的伴随法公式;并以算例形式,将伴随法计算的应力敏度结果与差分法结果进行对比,验证了所推导公式的准确性,应力敏度分析结果表明了应力对设计变量的偏导数具有局部性特点。在此基础上,以受应力约束重量极小化为目标的结构拓扑优化为例,对比分析了应力一阶Taylor近似与满应力法的优化效果。结果表明:相比满应力法,应力一阶近似能使结构应力在更多的部分达到许用应力,得到的最优结构重量更轻。对设计变量数目巨大的应力约束连续体结构拓扑优化问题,由于应力约束数目可以通过准有效约束初选及不考虑删除单元的应力约束等方式减少,通常比设计变量数目小很多,应用应力敏度分析伴随法可以显著提高计算效率。

For the topology optimization problems of continuum structures with stress constraints, an adjoint method for the sensitivity analysis of stresses are derived. With an example, the results of the sensitivity analysis of the stresses by the presented adjoint method are compared with those of the finite difference method; and it verifies the accuracy of the formula of the adjoint method. Meanwhile, it is shown from the results of stress sensitivity analysis that the partial derivatives of stresses with design variables are local. On this basis, taking the structural topology optimization problem with minimizing weight as the objective and subject to stress constraints for example, the effects on the optimization are compared between the first-order Taylor approximation of the stress and the fullstress method. The results show that the first-order approximation of stresses can make the stresses of more parts of the structure reach the allowable stress, and the weight of the optimal structure is lighter than that of the fullstress method. The number of design variables for the topology optimization problems of continuum structures with stress constraints is usually very large. The number of stress constraints can be reduced by the selection of quasi-effective stress constraints and there is no need to consider those stress constraints of the deleting elements. Therefore, the number of stress constraints is usually smaller than that of the design variables, and the adjoint method of stress sensitivity analysis can significantly improve the computational efficiency.