This work presents a rigorous mathematical formulation for topology optimization of a macrostructure undergoing ductile failure. The prediction of ductile solid materials which exhibit dominant plastic deformation is an intriguingly challenging task and plays an extremely important role in various engineering applications. Here, we rely on the phase-field approach to fracture which is a widely adopted framework for modeling and computing the fracture failure phenomena in solids. The first objective is to optimize the topology of the structure in order to minimize its mass, while accounting for structural damage. To do so, the topological phase transition function (between solid and void phases) is introduced, thus resulting in an extension of all the governing equations. Our second objective is to additionally enhance the fracture resistance of the structure. Accordingly, two different formulations are proposed. One requires only the residual force vector of the deformation field as a constraint, while in the second formulation, the residual force vector of the deformation and phase-field fracture simultaneously have been imposed. An incremental minimization principles for a class of gradient-type dissipative materials are used to derive the governing equations. Here, the level-set-based topology optimization is employed to seek an optimal layout with smooth and clear boundaries. Sensitivities are derived using the analytical gradient-based adjoint method to update the level-set surface for both formulations. Here, the evolution of the level-set surface is realized by the reaction-diffusion equation to maximize the strain energy of the structure while a certain volume of design domain is prescribed. Several three-dimensional numerical examples are presented to substantiate our algorithmic developments.

翻译：本文提出了一个严谨的数学框架，用于对经历韧性破坏的宏观结构进行拓扑优化。预测具有显著塑性变形的韧性固体材料是一项极具挑战性的任务，并在各类工程应用中扮演着至关重要的角色。研究中，我们采用相场断裂方法——一种被广泛用于模拟和计算固体断裂失效现象的框架。首要目标是在考虑结构损伤的前提下，优化结构拓扑以最小化其质量。为此，我们引入了拓扑相变函数（介于固体相与空隙相之间），从而扩展了所有控制方程。第二个目标是进一步增强结构的断裂抗力。据此，本文提出了两种不同的公式：一种仅需将变形场的残余力向量作为约束条件，而第二种公式则同时施加了变形场与相场断裂的残余力向量约束。基于一类梯度型耗散材料的增量最小化原理推导了控制方程。本文采用基于水平集的拓扑优化方法，寻求具有平滑清晰边界的优化布局。通过解析梯度伴随法推导灵敏度，以更新两种公式下的水平集曲面。其中，水平集曲面的演化通过反应-扩散方程实现，旨在最大化结构的应变能，同时预设设计域的一定体积约束。文中给出了多个三维数值算例，以验证所提算法的有效性。