This study presents a unified formulation of topology optimization with a finite strain nonlocal damage model using the continuous adjoint method. For the primal problem to describe the material response including deterioration, we consider the standard Neo–Hookean constitutive model and incorporate crack phase-field theory for brittle fracture within the finite strain framework. For the optimization problem, the objective function is set to accommodate multiple objectives by weighting each sub-function, and the continuous adjoint method is employed to derive the sensitivity. Thus, both the governing equations for primal and adjoint problems are written as strong forms and hold at any moment and at any location in the continuum body or on its boundary. Accordingly, the proposed formulation is independent of the requirements from numerical implementation, such as element type or discretization method. In addition, the reaction–diffusion equation is used to update the design variable in an optimizing process, by which the continuous distribution of the design variable, as well as material properties, are realized. After the basic performance of the proposed formulation is demonstrated with a simple numerical setup, two-material (matrix and inclusion materials) and single-material (material and null) topology optimizations are presented, and discussions are made.

中文翻译：

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使用连续伴随方法的有限应变非局部损伤模型的拓扑优化


本研究提出了一种使用连续伴随方法的有限应变非局部损伤模型的拓扑优化的统一公式。为了使原始问题描述包括劣化在内的材料响应，我们考虑了标准的 Neo-Hookean 本构模型，并在有限应变框架内纳入了脆性断裂的裂纹相场理论。对于优化问题，通过对每个子函数进行加权，将目标函数设置为容纳多个目标，并采用连续伴随法推导灵敏度。因此，主伴随问题和伴随问题的控制方程都写成强形式，并且在连续体体或其边界上的任何时刻和任何位置都成立。因此，所提出的公式独立于数值实现的要求，例如单元类型或离散化方法。此外，反应-扩散方程用于在优化过程中更新设计变量，从而实现设计变量的连续分布以及材料属性。在通过简单的数值设置演示了所提出的公式的基本性能之后，提出了两种材料（基体和夹杂物材料）和单一材料（材料和零材料）拓扑优化，并进行了讨论。