In this study, a non-probabilistic reliability-based topological optimization (NRBTO) method is investigated for the optimized design of the band-gap structure of two-dimensional phononic crystals (PnCs), in which unknown but bounded (UBB) uncertainty issues from the material uncertainty are taken into account. A measurement metric is first defined based on the interval set-theoretical model, which may transform the band-gap setting into a non-probabilistic reliability format. The interval series expansion and adjoint vector methods are then utilized to obtain design sensitivity expressions between frequency measures and design variables. The method of moving asymptote (MMA) is adopted as well to iteratively solve the reliability-oriented band-gap optimization problem for PnCs. To effectively tackle the gray-scale units, a mapping conversion method is involved eventually. The usage and validity of the present methodology are verified through three examples, and numerical results show that considering the uncertainty effect of UBB throughout the topology optimization process has an important impact on the final layout scheme. Additionally, the optimization of the out-of-plane modes of PnCs (both the band-gap upper bound and the filling rate constraint are considered) indicates that the topological configuration may always change accompanied with the filling rate increases.

在这项研究中，研究了一种基于非概率可靠性的拓扑优化（NRBTO）方法，用于二维声子晶体（PnC）的带隙结构的优化设计，其中存在未知但有界的（UBB）不确定性问题。考虑到材料的不确定性。首先基于间隔集理论模型定义测量度量，该模型可以将带隙设置转换为非概率可靠性格式。然后利用区间序列展开和伴随向量法来获得频率量度和设计变量之间的设计灵敏度表达式。还采用移动渐近线（MMA）方法来迭代地解决面向可靠性的PnC的带隙优化问题。为了有效应对灰度单位，最终涉及映射转换方法。通过三个实例验证了本方法的有效性和有效性，数值结果表明，在整个拓扑优化过程中考虑UBB的不确定性影响对最终布局方案具有重要影响。此外，对PnC平面外模式的优化（考虑了带隙上限和填充率约束）表明拓扑结构可能总是随着填充率的增加而改变。