基于固定网格和拓扑导数的结构拓扑优化自适应泡泡法

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.