空间刚架的拓扑-截面形状协同优化
COLLABORATIVE OPTIMIZATION OF TOPOLOGY AND CROSS-SECTIONAL SHAPE FOR SPACE FRAME STRUCTURES
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摘要: 刚架结构常见于建筑学、土木工程等领域, 其设计通常伴随着结构减重与力学性能提升的双重需求. 为实现以上设计目标, 结构优化方法被广泛采纳, 并用于解决刚架结构的形状与拓扑优化问题. 基结构法是研究离散桁架或刚架结构优化的基本方法, 在此基础上, 节点移动基结构法被提出并有效扩展了传统方法的设计空间. 然而, 当前节点移动基结构法的相关研究均基于圆形梁截面, 并未考虑梁截面形状的优化以及不规则截面形状引起的主方向设计问题. 因此, 文章基于罗德里格斯旋转以及离散截面形状优化方法, 提出了一种空间刚架的拓扑-截面形状协同优化方法. 该方法解决了节点移动引起的梁截面主方向自适应更新问题, 通过引入转角设计变量扩展了节点移动基结构法这一设计框架. 本方法还考虑了不同截面形状梁在抗弯与抗扭性能方面的差异, 可实现多种梁截面形状的自动选择, 从而保证复杂工况下材料的有效利用. 数值算例表明, 通过结构拓扑与截面形状的协同优化, 可以获得性能更为优异的空间刚架构型.Abstract: Frame structures are commonly used in architecture, civil engineering and other fields, and their design often necessitates a balance between weight reduction and the enhancement of mechanical properties. To achieve these design objectives, structural optimization methods are widely used to solve the shape and topology optimization problems of frame structures. The ground structure method is a fundamental approach for optimizing discrete truss or frame structures. The traditional ground structure method employs a set of nodes with fixed positions and achieves structural topology optimization by modifying the connectivity of members between the nodes. The introduction of the node-moving ground structure method has effectively expanded the design space of the traditional methods. However, current research on the node-moving ground structure method mostly is based on circular beam cross-sections, without considering the optimization of cross-sectional shapes or the principal direction design problems arising from irregular cross-sectional shapes. This limitation restricts the design space of existing optimization methods. Therefore, this paper proposes a collaborative optimization method for topology and cross-sectional shape of space frame structures, based on Rodrigues’ rotation rule and discrete cross-sectional shape optimization. This method addresses the issue of adaptive updating of the principal direction of beam cross-sections caused by node movement. By utilizing Rodrigues’ rotation, it establishes a unique correspondence between the principal direction of the initial structure and that of the structure after node movement. On this basis, the design variable of cross-sectional rotation angle is introduced, effectively expanding the design framework of the node-moving ground structure method. The method also takes into account the differences in bending and torsional resistance among beams with different cross-sectional shapes, enabling the automatic selection of various beam cross-sectional shapes to ensure efficient material utilization under complex conditions. Numerical examples demonstrate that through the collaborative optimization of structural topology and cross-sectional shape, frame configurations in three-dimensional space with superior performance can be obtained.