基于固定网格和拓扑导数的结构拓扑优化自适应泡泡法

ADAPTIVE BUBBLE METHOD USING FIXED MESH AND TOPOLOGICAL DERIVATIVE FOR STRUCTURAL TOPOLOGY OPTIMIZATION

为继承传统拓扑优化泡泡法变量少、精度高等优点,并克服其网格重划频繁、孔洞合并操作繁琐等不足,提出了一种基于固定网格和拓扑导数的自适应泡泡方法.该方法的主要特点是:(1)采用有限胞元固定网格分析方法计算结构力学响应,在优化过程中无需网格更新和重划分,就能保证较高的分析精度;(2)根据拓扑导数信息指导结构区域中孔洞的引入,不仅消除了优化结果对孔洞初始布局的依赖性,还能有效控制设计变量的数量;(3)引入拓扑导数阈值和孔洞影响区域新概念,实现了孔洞引入频次和位置的自适应调节,保证了拓扑优化过程的数值计算稳定性;(4)采用光滑变形隐式曲线描述孔洞边界,不仅设计参数少、变形能力强,而且便于处理孔洞间的融合/分离操作以及与固定网格分析方法的有机结合.理论分析和数值算例表明,改进后的自适应泡泡法能够消除传统泡泡法因采用拉格朗日网格和参数化B样条曲线模型而存在的实施困难,采用很少的设计变量就可获得边界光滑清晰的优化结果.

In this paper, an improved topology optimization approach named adaptive bubble method (ABM) is proposed to overcome the shortcomings of the traditional bubble method, such as the frequent remeshing operation and the tedious merge process of holes. The main characteristics of ABM are summarized as follows:(1) The finite cell method (FCM) is adopted to perform high-precision numerical analysis within the fixed Eulerian mesh, so that the processes of mesh updating and remeshing are no longer needed;(2) The topological derivative is calculated for the iterative position of new holes into the design domain, which can completely solve the initial layout dependency problem and significantly reduce the number of design variables;(3) New concepts related to the topological derivative threshold and the influence region of inserted holes are defined to adaptively adjust the inserting frequency and inserting position of new holes, and the numerical stability of topology optimization could then be kept very well;(4) The smoothly deformable implicit curve (SDIC), which is characterized by very few parameters and high deformation capacity, is utilized to describe the hole boundary, since SDIC could facilitate the fixed-grid analysis as well as the merge process of holes. The structural optimization based on ABM is essentially a collaborative design process that contains the shape optimization of inserted holes as well as the topology changes caused by the insertion of new holes and the merging/separation of inserted holes. Theoretical analysis and numerical results showed that ABM can be implemented conveniently thanks to the adoption of the FCM/SDIC framework, and the optimized results featured by clear and smooth boundaries could be obtained with much less number of design variables by using ABM. Namely, the proposed ABM retains all the advantages of the traditional bubble method, while effectively breaking through its development bottleneck caused by the use of lagrangian description and the parametric B-spline curve.