频带约束下刚架结构优化问题可行域研究

研究频带约束下刚架结构轻量化设计问题的可行域基本性质.刚架结构中梁的断面积取为设计变量,采用欧拉-伯努利梁的动力刚度法及W-W算法精确求解结构的固有频率值.利用W-W算法的特征值计数原理,研究了前述优化问题的可行域的形状和联通性,发现可行域呈现复杂的形状,由多块非联通子域组成,并且部分可行域子域可以是低维的.还以三杆梁的尺寸优化和拓扑优化为例,给出了可行域的具体形状,展示了这一优化问题的可行域具有"强奇异性".可行域的这一特点给基于梯度类的优化算法带来了极大的困难,需要采用其他手段处理,也使这类问题有望用于测试各类优化软件算法.

Research on feasible domain of optimization problem of rigid frame structures with frequency band constraints

The feasible domain of minimum weight designs of rigid frame structures with frequency band constraints is studied, in which the cross sectional area of beam is chosen as design variable. The natural frequencies of frame structures are solved with the dynamic stiffness method and W-W algorithm for Euler-Bernoulli beam. Then, the shape and connectivity of the feasible domain of the optimization problem are described exactly by eigenvalue counting technology. It is found that the feasible domain of the optimization problem is complex and disjointed, and some disjointed domains are low dimensional. The shape and connectivity of the feasible domain of the size and topology optimization problem of 3-beam model are studied, and results show that the singularity is very strong, which brings about great difficulties for the process of classical gradient-based optimization searching algorithm, and other way is needed. It also provides a bench mark problem for the test of soft optimum algorithm.