This study presents a unified formulation of topology optimization for finite strain elastoplastic materials. As the primal problem to describe the elastoplastic behavior, we consider the standard -plasticity model incorporated into Neo-Hookean elasticity within the finite strain framework. For the optimization problem, the objective function is set to accommodate both single and multiple objectives, the latter of which is realized by weighting each sub-function. The continuous adjoint method is employed to derive the sensitivity, which is a general form that accepts any kind of discretization method. Then, the governing equations of the adjoint problem are derived as a format that holds at any moment and at any location in the continuum body or on its boundary. Accordingly, the proposed formulation is independent of any requirements in numerical implementation. In addition, the reaction–diffusion equation is used to update the design variable in an optimizing process, for which the continuous distribution of the design variable as well as material properties are maintained. Two specific optimization problems, stiffness maximization and plastic hardening maximization, for two and three-dimensional structures are presented to demonstrate the ability of the proposed formulation.

中文翻译：

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使用连续伴随法的有限应变弹塑性材料拓扑优化：公式、实现和应用


这项研究提出了有限应变弹塑性材料拓扑优化的统一公式。作为描述弹塑性行为的首要问题，我们考虑将标准塑性模型纳入有限应变框架内的新胡克弹性。对于优化问题，目标函数设置为适应单个目标和多个目标，后者通过对每个子函数进行加权来实现。采用连续伴随法来推导灵敏度，这是接受任何类型离散化方法的通用形式。然后，将伴随问题的控制方程导出为在连续体或其边界上的任何时刻和任何位置都成立的格式。因此，所提出的公式独立于数值实现中的任何要求。此外，反应扩散方程用于在优化过程中更新设计变量，从而保持设计变量和材料性能的连续分布。提出了二维和三维结构的两个具体优化问题，即刚度最大化和塑性硬化最大化，以证明所提出的公式的能力。