TOPOLOGY OPTIMIZATION OF THIN SHELL STRUCTURES BASED ON ADAPTIVE BUBBLE METHOD
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摘要: 为有效解决薄壳结构拓扑优化设计难题, 并满足其对分析模型精度和优化结果质量的高要求, 结合等几何壳体分析方法提出一种基于自适应泡泡法的新型拓扑优化设计框架. 等几何分析技术在薄壳分析方面具有天然的优势: 一方面可为薄壳结构建立起精确的NURBS分析模型, 避免了模型转换操作及误差; 另一方面还可保证待分析物理场的高阶连续性, 无需设置转角自由度等. 为了在给定壳面上实现结构的拓扑演化, 借助NURBS曲面(即等几何分析中的薄壳中面)的映射关系, 仅需在规则的二维参数区域内改变结构拓扑即可. 鉴于此, 采用自适应泡泡法在壳面参数区域内开展拓扑优化, 该方法包含孔洞建模、孔洞引入和固定网格分析3个模块, 其在当前工作中分别基于闭合B样条、拓扑导数理论和有限胞元法实现. 其中, 闭合B样条兼具参数和隐式两种表达形式, 参数形式便于在CAD系统中直接生成精确的结构模型; 隐式形式不仅便于开展孔洞的融合/分离操作, 还能与有限胞元法有机结合以替代繁琐的修剪曲面分析方法. 理论分析和数值算例表明, 所提优化设计框架将复杂的薄壳结构拓扑优化问题转化为简单的二维结构拓扑优化问题, 在保证足够分析精度的基础上使用相对很少的设计变量就可得到具有清晰光滑边界且便于导入到CAD系统的优化结果.Abstract: Combined with the isogeometric shell analysis (IGA) method, a new optimization design framework based on the adaptive bubble method (ABM) is proposed in this work, in order to effectively solve the topology optimization design problem of thin shell structures, and meet the high standards for the accuracy of analysis models as well as the quality of optimization results. The IGA technique has its natural advantages in thin shell analysis: on one hand, precise analysis models for thin shell structures are established with IGA, and model transformation operations and the resulting errors could therefore be avoided; on the other hand, the high-order continuity of physical fields to be solved can be guaranteed without setting the rotational degrees of freedom. Thanks to the mapping relationship related to the NURBS surface (i.e., the middle surface of thin shell), the structural topology evolution of a given shell surface can be achieved easily in the 2D regular parametric domain. In view of this, the ABM is adopted to carry out topology optimization in the parametric domain, and it contains three modules: the modeling of holes with closed B-splines (CBS), the insertion of holes via the topological derivative theory, and the fixed-grid analysis based on the finite cell method (FCM). It should be noted that holes are expressed in both parametric and implicit forms with CBS. The parametric form makes it convenient to import the structural model into the CAD system exactly. The implicit form not only facilitates the merging and separating operations of holes, but also can be well combined with the FCM which is far more convenient than trimming surface analysis (TSA). Theoretical analysis and numerical examples indicate that the proposed design framework could convert the complicated thin shell structural topology optimization problem into the simple one in the 2D domain, and optimized results with clear and smooth boundaries can be obtained with relatively few design variables.
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Key words:
- topology optimization /
- thin shell /
- adaptive bubble method /
- isogeometric analysis /
- closed B-splines
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