A sequentially coupled shape and topology optimization framework is presented in which the outer geometry and the internal topological layout of beam-type structures are optimized simultaneously. The outer geometry of the beam-type structures is parametrically described by non-uniform rational B-splines (NURBS), which guarantees a highly accurate description of the structural shape and enable an efficient control of the design domain with only a few control points. The computational efficiency of the coupled optimization approach is assured by applying a gradient-based optimization algorithm, for which the sensitivities are derived in closed form. The formulation of the coupled optimization approach is tailored toward 2.5D and full 3D representations of beam structures used in engineering applications. The 2.5D beam model, which has been taken from the literature, uses standard beam elements to simulate the beam response in the longitudinal direction, whereby the cross-sectional properties of the beam elements are calculated from additional 2D finite element method (FEM) analyses. A comparison study of a cantilever beam problem subjected to pure shape optimization and pure topology optimization illustrates that the 2.5D and 3D beam models lead to similar shape and topology designs, but that the 2.5D beam model has a significantly higher computational efficiency. Specifically, the computational times for the 2.5D model are about a factor 70 (shape optimization) and 1.4 (topology optimization) lower than for the 3D model, which indicates that in the coupled optimization approach the optimization of the shape provides the largest contribution to the higher computational efficiency of the 2.5D model. The coupled shape and topology optimization analysis subsequently performed on the 2.5D cantilever beam model demonstrates that the specific order at which the alternating shape and topology optimization increments are performed in the staggered update procedure turns out to have some influence on the computational speed and the value of the minimal compliance computed. Despite these differences, the final beam structures following from the different staggered update procedures illustrate how shape and topology can be efficiently optimized in an integrated, coupled fashion.

提出了一种顺序耦合的形状和拓扑优化框架，其中梁型结构的外部几何形状和内部拓扑布局被同时优化。梁型结构的外部几何形状由非均匀有理B样条（NURBS）参数化描述，这保证了对结构形状的高度精确描述，并且仅需几个控制点就可以有效控制设计域。耦合优化方法的计算效率通过应用基于梯度的优化算法来保证，对于该算法，敏感度以封闭形式导出。耦合优化方法的制定是针对工程应用中使用的梁结构的2.5D和完整3D表示量身定制的。2.5D光束模型，从文献中获得的结果是使用标准梁单元来模拟纵向方向上的梁响应，从而通过附加的2D有限元方法（FEM）分析来计算梁单元的截面特性。对经过纯形状优化和纯拓扑优化的悬臂梁问题的比较研究表明，2.5D和3D光束模型导致相似的形状和拓扑设计，但是2.5D光束模型具有明显更高的计算效率。具体来说，2.5D模型的计算时间比3D模型的计算时间低约70倍（形状优化）和1.4倍（拓扑优化），这表明在耦合优化方法中，形状的优化为2.5D模型的更高计算效率提供了最大的贡献。随后在2.5D悬臂梁模型上进行的形状和拓扑优化耦合分析表明，交错更新过程中形状和拓扑优化增量交替执行的特定顺序对计算速度和值有一定影响。计算出的最小遵从性。尽管存在这些差异，但来自不同交错更新过程的最终波束结构说明了如何以集成，耦合的方式有效地优化形状和拓扑。随后在2.5D悬臂梁模型上进行的形状和拓扑优化耦合分析表明，交错更新过程中形状和拓扑优化增量交替执行的特定顺序对计算速度和值有一定影响。计算出的最小遵从性。尽管存在这些差异，但来自不同交错更新过程的最终波束结构说明了如何以集成，耦合的方式有效地优化形状和拓扑。随后在2.5D悬臂梁模型上进行的形状和拓扑优化耦合分析表明，交错更新过程中形状和拓扑优化增量交替执行的特定顺序对计算速度和值有一定影响。计算出的最小遵从性。尽管存在这些差异，但来自不同交错更新过程的最终波束结构说明了如何以集成，耦合的方式有效地优化形状和拓扑。5D悬臂梁模型表明，在交错更新过程中执行交替形状和拓扑优化增量的特定顺序实际上会对计算速度和所计算的最小依从性值产生一定影响。尽管存在这些差异，但来自不同交错更新过程的最终波束结构说明了如何以集成，耦合的方式有效地优化形状和拓扑。5D悬臂梁模型表明，在交错更新过程中执行交替形状和拓扑优化增量的特定顺序实际上会对计算速度和所计算的最小依从性值产生一定影响。尽管存在这些差异，但来自不同交错更新过程的最终波束结构说明了如何以集成，耦合的方式有效地优化形状和拓扑。