We propose a fully immersed topology optimization procedure to design structures with tailored fracture resistance under linear elastic fracture mechanics assumptions for brittle materials. We use a level set function discretized by radial basis functions to represent the topology and the Interface-enriched Generalized Finite Element Method (IGFEM) to obtain an accurate structural response. The technique assumes that cracks can nucleate at right angles from the boundary, at the location of enriched nodes that are added to enhance the finite element approximation. Instead of performing multiple finite element analyses to evaluate the energy release rates (ERRs) of all potential cracks—a procedure that would be computationally intractable—we approximate them by means of topological derivatives after a single enriched finite element analysis of the uncracked domain. ERRs are then aggregated to construct the objective function, and the corresponding sensitivity formulation is derived analytically by means of an adjoint formulation. Several numerical examples demonstrate the technique’s ability to tailor fracture resistance, including the well-known benchmark L-shaped bracket and a multiple-loading optimization problem for obtaining a structure with fracture resistance anisotropy.

我们提出了一种完全浸入式拓扑优化程序，以在脆性材料的线性弹性断裂力学假设下设计具有定制断裂阻力的结构。我们使用由径向基函数离散化的水平集函数来表示拓扑，并使用界面丰富的广义有限元方法 (IGFEM) 来获得准确的结构响应。该技术假设裂纹可以从边界以直角成核，在增加的丰富节点的位置，以增强有限元近似。而不是执行多个有限元分析，以评估所有潜在的裂纹-一个过程的能量释放率（ERRS），这将是难以计算-我们通过拓扑衍生物的装置之后的近似它们单未破解域的丰富有限元分析。然后聚合 ERR 以构建目标函数，并通过伴随公式分析导出相应的灵敏度公式。几个数值示例证明了该技术能够调整抗裂性，包括众所周知的基准 L 形支架和用于获得具有抗裂性各向异性的结构的多载荷优化问题。