$\partial^2 ( \mathrm{TO} ) $：基于双拓扑导数的三维脆性固体断裂缓解增强拓扑优化
$\partial^2 ( \mathrm{TO} ) $: A Dual Topological Derivative-Based Enriched Topology Optimization for Fracture Mitigation in 3-D Brittle Solids
AI总结：
该研究针对三维脆性固体提出断裂缓解拓扑优化框架，用径向基函数参数化水平集函数描述拓扑，利用双拓扑导数及非局部应力恢复程序，通过p均值函数聚合边界ERRs，经三维数值示例验证了框架能力。
AI中文摘要：
我们提出了一种用于三维脆性固体的断裂缓解拓扑优化框架。拓扑由径向基函数参数化的水平集函数描述，结构响应使用界面增强有限元公式计算。双拓扑导数有两个作用。一是在优化过程中在固体内部形成空洞，二是仅通过对未开裂几何形状的单次增强有限元分析的应力场来评估整个边界上的能量释放率（ERRs）。为此，假设使用最大环向应力准则在沿边界引入的增强节点位置形成便士形裂纹以进行精确有限元分析。由于ERR估计对应力精度敏感，我们使用非局部应力恢复程序计算节点应力场。拓扑优化目标使用p均值函数聚合边界ERRs。包括常用的L形支架基准问题在内的三维数值示例证明了所提出框架的能力。
英文摘要：
We propose a fracture-mitigation topology optimization framework for 3-D brittle solids. The topology is described by a level set function parameterized by radial basis functions, and the structural response is computed using an interface-enriched finite element formulation. Dual topological derivatives serve two purposes. First, they are used to nucleate holes within the solid during the optimization process. Second, they are used to evaluate energy release rates (ERRs) along the entire boundary, requiring only the stress field from a single enriched finite element analysis of the uncracked geometry. For this purpose, penny-shaped cracks are assumed to nucleate using the maximum hoop stress criterion, at the locations of enriched nodes introduced along the boundary for accurate finite element analysis. Because ERR estimates depend sensitively on stress accuracy, we compute a nodal stress field using a non-local stress-recovery procedure. The topology optimization objective aggregates the boundary ERRs using a $p$-mean function. Three-dimensional numerical examples, including the commonly studied L-bracket benchmark problem, demonstrate the capability of the proposed framework.
