Abstract This paper considers topology optimization approaches where the geometry is described by a level set method and the system’s response is discretized on nonconforming meshes while providing a crisp definition of interface and external boundaries. Since the interface is explicitly tracked, the elements intersected by the interface are divided into sub-elements to which a phase needs to be assigned. Due to loss of information in the discretization of the level set field, certain geometrical configurations allow for ambiguous phase assignment of sub-elements, and thus ambiguous definition of the interface. The study presented here focuses on investigating these topological ambiguities in embedded geometries constructed from discretized level set fields on hexahedral meshes. Three-dimensional problems where several intersection configurations can significantly affect the problem’s topology are considered. This is in contrast to two-dimensional problems where ambiguous topological features exist only in one intersection configuration, and identifying and resolving them is straightforward. A set of rules that resolve these ambiguities for two-phase problems is proposed, and algorithms for their implementations are provided. The influence of these rules on the evolution of the geometry in the optimization process is investigated with linear elastic topology optimization problems. These problems are solved by an explicit level set topology optimization framework that uses the extended finite element method to predict physical responses. This study shows that the choice of a rule to resolve topological features can result in drastically different final geometries. However, for the problems studied in this paper, the performances of the optimized design do not differ.

中文翻译：

---

拓扑优化中离散化 3D 几何的模糊相位分配

摘要 本文考虑了拓扑优化方法，其中几何由水平集方法描述，系统的响应在不一致的网格上离散化，同时提供界面和外部边界的清晰定义。由于显式跟踪界面，因此与界面相交的元素被划分为需要分配阶段的子元素。由于水平集域离散化中的信息丢失，某些几何配置允许子元素的相位分配不明确，因此界面的定义不明确。此处介绍的研究侧重于研究由六面体网格上的离散化水平集场构建的嵌入式几何中的这些拓扑模糊性。考虑了多个交叉点配置可以显着影响问题拓扑的三维问题。这与二维问题相反，在二维问题中，模棱两可的拓扑特征仅存在于一个交集配置中，识别和解决它们很简单。提出了一组解决两阶段问题的歧义的规则，并提供了实现它们的算法。通过线弹性拓扑优化问题研究了这些规则对优化过程中几何演变的影响。这些问题由显式水平集拓扑优化框架解决，该框架使用扩展的有限元方法来预测物理响应。这项研究表明，选择解决拓扑特征的规则可能会导致最终几何形状截然不同。然而，对于本文所研究的问题，优化设计的性能没有差异。