桁架优化解存在性的研究

On Existence of Optimization Solution of Truss Structure

研究了桁架静力与动力学优化解的存在性 ,阐明了该研究的意义。由于除固频约束有界非凸外 ,所有静力与动力学约束均为有界凸约束 ,从而证明了 :任何构型确定的桁架 ,其拓扑与几何形状已定 ,当以各杆截面积为设计变量进行尺寸优化时 ,若各变量可连续变化且其上限足够大 ,则桁架在凸约束下的静力、动力学优化解存在。当桁架的固频约束已被满足 ,若再附加其它凸约束时 ,其动力学优化解存在。已经证明 ,上述桁架的尺寸优化虽能连续改变其固频数值 ,但改变范围不够大。固频并非总能在优化中得到满足 ,但改变桁架的拓扑构型 ,固频约束可行域会有较大变化 ,原本不能满足的固频约束可能得到满足。因此 。

In our extensive research experience on optimization solution of truss structures, we actually had the unpleasant experience of not being able to find an optimal solution of a certain truss structure no matter how we tried. We presented existence theory on optimization solution of engineering structures in a previous paper . Armed with such existence theory, we are confident that we can find optimal solutions if conditions stipulated in existence theory can be met. In this paper, we present and give proofs of six theorems about existence of optimal solution of truss structure. We now give a general explanation of these six theorems without going into details. All the constraints, except natural frequency which is non-convex but bounded, are convex constraints and bounded. For a truss whose topology and geometric configuration are unvaried, if the cross-sectional areas of bars can vary continuously and their upper bounds are large enough, optimal solution exists under either static or dynamic convex constraints. For such a truss, if the required natural frequencies can be satisfied, the dynamic optimal solution exists under additional convex constraints. Sizing optimization can change the feasible domain of natural frequency constraint only in a small range for truss with a determinate construction scheme. Only the feasible domain of constraint of natural frequency is non-convex and the constraint of natural frequency cannot always be satisfied. However, the feasible domain of constraint of natural frequency will be changed and followed by the change of topology configuration. Consequently, unsatisfied natural frequency can be satisfied if the topology configuration changes in some way. If p is the required p th frequency and if, for a certain truss of effective topology group, l p p u p, where superscripts u and l are the upper and lower bound of p th natural frequency respectively, then the optimization solution exists. Conversely, if, for all effective topology groups, the above mentioned inequality cannot be met then the optimization solution will not exist.